An algorithm for distributed time delay identification without a priori knowledge of the kernel

Time delays are ubiquitous in industry and nature, and they significantly affect both transient dynamics and stability properties. Consequently, it is often necessary to identify and account for the delays when, e.g., designing a model-based control strategy. However, identifying delays in differential equations is not straightforward and requires specialized methods. Furthermore, existing approaches for identifying distributed time delays in delay differential equations (DDEs) require that the functional form of the involved kernel (also called memory function) is known. In this work, we propose an algorithm for identifying distributed delays, which does not require a priori knowledge of the kernel. Specifically, we 1) approximate the unknown kernel in the DDEs by the probability density function of a mixed Erlang distribution and 2) use the linear chain trick (LCT) to transform the resulting DDEs into ODEs. Finally, the parameters in the kernel approximation are estimated as the solution to a dynamical maximum likelihood problem, and we use a single-shooting approach to approximate this solution. We demonstrate the efficacy of the algorithm using a numerical example that involves a point reactor kinetics model of a molten salt nuclear fission reactor.